Phantom bursters were introduced to explain bursting electrical activity in β-cells with different periods. We study a polynomial version of the phantom bursting model. In particular we analyse the fast subsystem, where the slowest variable is assumed constant. We find the equilibrium points of the fast subsystem and analyse their stability. Furthermore an analytical analysis of the existence of Hopf bifurcation points and the stability of the resulting periodics is performed by studying the sign of the first Lyapunov coefficient.